Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to with scale factor . Justify the construction. Are the two triangles congruent? Note that all the three angles and two sides of the two triangles are equal
Solution
Steps of construction
Draw a circle of radius 4 cm. Construct a pair of tangents to it, the angle between which is . Also justify the
construction. Measure the distance between the centre of the circle and the point of intersection of tangents
Solution
Steps of construction
1. Draw a circle of radius OA = 4 cm with centre O
2. Produce OA to P such that
3. Draw a perpendicular bisector of OP=8cm
4.Now taking A as Centre draw circle of radius AP = OA = 4 cm
5.Which intersect the circle at x and y
6.Join PX and PY
7.PX and PY is the tangent of the circle
Justification
In we have
(Radius)
(Radius of circle with centre A)
is equilateral triangle
In we have
Hence
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Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and .Construct a triangle similar to ABC with scale factor . Justify the construction
Solution
Given : AB = 5 cm, BC = 6 cm
Steps of construction
Justification: Here, [by construction]
Now
Also,
Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5 cm. Construct a triangle PQR similar to ABC in which PQ = 8 cm. Also justify the Construction.
Solution
Steps of construction
1. Draw line BC = 5 cm
2. Taking B and C as centres, draw two arcs of equal radius 6 cm intersecting each other at point A.
3. Join AB and AC DABC is required isosceles triangle
4 From B draw ray with an acute angle
6. draw at with equal distance
7. Join and from draw line , intersect extended segment BC at point D.
8. From point D draw meting BA produced at E.
Then EBD is required triangle. We can name it PQR.
Justification
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Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.
Solution
Steps of construction
1. Draw two concentric circles with center O and radii 3 cm and 5 cm
2. Taking any point P on outer circle, Join P and O
3. Draw perpendicular bisector of OP let M be the mid point of OP
4. Taking M as centre and OM as radius draw a circle which cuts inner circle at Q and R
5. Join PQ and PR. Thus PQ and PR are required tangents
On measuring PQ and PR we find that PQ = PR = 4 cm
Calculations
[using pythagoras theorem]
Hence the length of both tangents is 4 cm.
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Draw a parallelogram ABCD in which BC = 5 cm, AB = 3 cm and , divide it into triangles BCD and ABD
by the diagonal BD. Construct the triangle BD'C' similar to DBDC with scale factor . Draw the line segment D'A' parallel to DA where A' lies on extended side BA. Is A'BC'D' a parallelogram?
Solution
Steps of construction
1. Draw, A line AB = 3 cm
2 Draw a ray by making
3. Taking B as centre and radius equal to 5 cm. Draw an arc which cut BP at point C
4. Again draw ray AX making
5. With A as centre and radius equal to 5 cm draw an arc which cut AX at point D
6. Join C and D Here ABCD is a parallelogram
7. Join BD , BD is a diagonal of parallelogram ABCD
8. From B draw a ray BQ with any acute angle at point B i.e., is acute angle
9. Locate 4 points on BQ with equal distance.
10. Join and from parallel to which intersect at point
11. From point draw line which is parallel to CD
12. Now draw a line segment parallel to DA
Note : Here and are the extended sides.
13. is a parallelogram in which and and divide it into triangles and by the diagonal
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Two line segments AB and AC include an angle of where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that and . Join P and Q and measure the length PQ.
Solution
Given :
AB = 5 cm and AC = 7 cm
From equation 1
P is any point on B
scale of a line segment AB is
Steps of construction
1. Draw line segment AB = 5 cm
2. Now draw ray AO which makes an angle i.e.,
3. Which A as center and radius equal to 7 cm draw an arc cutting line AO at C
4. Draw ray AP with acute angle BAP
5. Along AP make 4 points with equal distance.
6. Join
7. From draw which is parallel to which meet AB at point P.
Then P is point which divides AB in ratio 3 : 1
AP : PB = 3 : 1
8. Now draw ray AQ, with an acute angle CAQ.
9. Along AQ mark 4 points with equal distance.
10. Join
11. From draw which is parallel to which meet AC at point Q.
Then Q is point which divides AC in ratio 1 : 3
AQ : QC = 1 : 3
12. Finally join PQ and its measurement is 3.25 cm.
Construct a tangent to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre
Solution
Given : Radius = 4cm
Steps of construction
1. Draw a circle at radius r = 4 cm at point O.
2. Take a point P at a distance 6cm from point O and join PO
3. Draw a perpendicular bisector of line PO M is the mid-point of PO.
4. Taking M as centre draw another circle of radius equal to MO and it intersect the given circle at point Q and R
5. Now, join PQ and PR.
Then PQ and PR one the required two tangents.
Draw a triangle ABC in which BC = 6 cm, CA = 5 cm and AB = 4 cm.Construct a triangle similar to it and of scale factor
Solution
Given : BC = 6cm, CA = 5cm, AB = 4cm
Scale factor = 5/3
Let m = 5 and n = 3
Steps of construction
7. Draw a line through parallel to AC intersect AB extended at .
Now is required triangle.
View Full Answer(1)Draw a right triangle ABC in which BC = 12 cm, AB = 5 cm and .Construct a triangle similar to it and of scale factor . Is the new triangle also a right triangle ?
Solution
Given BC = 12cm, AB = 5 cm,
Steps of construction
1. Draw a line BC = 12 cm
2. From B draw AB = 5 cm which makes an angle of at B.
3. Join AC
4. Make an acute angle at B as
5. On BX mark 3 point at equal distance at .
6. Join
7. From draw intersect AB at
8. From point draw intersect AB
Now is the required triangle and is also a right triangle.
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